Discrete Mathematics is the study of mathematics limited to a set of integers. Discrete Mathematics is becoming the basis of many real-world problems, particularly in computer science. From our daily experience, we can say that natural languages are not accurate as they can have a different meaning. They are ambiguous and not suitable for coding purposes. Therefore we develop a formal language called the object language. In this language, we use a well-defined object followed by a definite statement regarding the same object. When we use mathematical expressions to denote the logical statements, we call this Discrete Mathematics, also commonly paired with Graph Theory. Discrete Mathematics is gaining popularity these days because of it’s increasing usage in computer science. Complex logic and calculations can be depicted in the form of simple statements. It is used in daily life in the following ways:

1.) Algorithms All of us write codes on the computer on some platform with built-in languages like C, Python, Java etc. but before writing the codes itself we prefer writing the algorithms, which involves basic logic for the code using discrete maths. A computer programmer uses discrete math to design efficient algorithms. This design includes discrete math applied to determine the number of steps an algorithm needs to complete, which implies the speed of the algorithm. Algorithms are the rules by which a computer operates. These rules are created through the laws of discrete mathematics. Because of discrete mathematical applications in algorithms, nowadays computers run faster than ever before. Example of an algorithm: procedure multiply(a , b: positive integers) {the binary expansions of a and b are ( ) and ( ) respectively for j=0 to j=n-1 if then shifted j places else 0 { } p=0 for j=0 to j=n-1 p = p + return p {p is the value of ab} We can clearly see the application of logic and Discrete maths in the above algorithm.

2.) Cryptography The field of cryptography is based entirely on discrete mathematics. Cryptography is the study of how to create security structures and passwords for computers and other electronic systems. One of the most important parts of discrete mathematics is Number theory which allows cryptographers to create and break numerical passwords. Because of the amount of money and the amount of confidential information involved, cryptographers must first have a solid background in number theory to show they can provide secure passwords and encryption methods. Shown below is an example of Discrete Mathematics in encryption.

3.) Computer Programmes The tasks running on computer use one or another form of discrete maths. The computer functions in a specific way depending on the decisions made by the user.

For example: Discrete Mathematics is very closely connected with Computer Science. Theoretical Computer Science, the foundation of our field is often considered a subfield of discrete mathematics. Computer Science is built upon logic, and numerous, if not most, areas of discrete mathematics utilized in the field. For Example: p(x) denote “number x+4 is an even integer” ~p(x) denote “number x+4 is not an even integer” q(x,y) to represent an open statement that contains 2 variables. With p(x) and q(x,y) as above, universe still concern itself with integers only, make replacements for x,y we get: p(5) = (5+2) is an even integer ~p(7) = (7+2) is not an even integer q(4,2) = numbers 4,2,8 are even integers “For some x” and “For some x,y” are said to quantify the open statement p(x) and q(x,y) respectively

- For some x, p(x)
- For some x,y q(x,y)

4.) Computing Rankings Discrete mathematics describe processes that consist of a sequence of individual steps. Many ways of producing rankings use both discrete maths and graph theory. Specific examples include the ranking relevance of search results using Google, ranking teams for tournaments or chicken pecking orders, and ranking sports team performances or restaurant preferences that include apparent paradoxes.

5.) Train Delay Discrete mathematics is used in a really new way in the UK. Discrete math is used in choosing the most on-time route for a given train trip. The software is under development and uses discrete math to calculate the most time efficient route for a passenger. Each change of train by a passenger at a station is like an obstacle because of possible delays, spreads out the arrival time of the passenger at the next station on the route. For every part of the journey, the kernel for each station is applied in succession, giving the distribution of arrival time at the final destination. Working of the system:- Each station has a 60 x 60 matrix for a particular time of day. It is 60 on one side because the maximum delay considered is an hour. On the other side it is 60 because the hour is divided up into discrete one-minute intervals, the nearest value provided by the train timetables.

The matrix is fitted with the probability that if you arrive at the station at the minute I, you depart at minute j. This is based on timetable information and the delay profile information obtained from the website data grab. The matrices for each station are in turn applied to a column vector. The column vector contains the probability distribution of your arrival time at the next station with each value showing the probability of being 0, 1,2, 3 minutes late etc. The total column vector sums to one. Before you depart, the first value in the column vector is 1 and the rest are zeros – a delta function. This is because you haven’t had a chance to be subjected to delays yet. By applying your starting station’s matrix to this column vector, a new one is generated containing the probability distribution of your arrival time at the next station. The matrix for that station is then applied to the new column vector, and so on until you reach your destination. The final, resultant column vector provides the distribution of your probable arrival times. This can then be compared with the final column vector for other routes and the optimum route selected. A railway control office using Mathematics and Graphs to analyze patterns.

6.) Airplane Deviation Graphs are nothing but connected nodes(vertex). So any network related, routing, finding a relation, path etc related real-life applications use graphs. Aircraft scheduling: Assuming that there are k aircraft and they have to be assigned n flights. The ith flight should be during the time interval (ai, bi). If two flights overlap, then the same aircraft cannot be assigned to both the flights. This problem is modeled as a graph as follows. The vertices of the graph correspond to the flights. Two vertices will be connected if the corresponding time intervals overlap. Therefore, the graph is an interval graph that can be colored optimally in polynomial time. Below is an example of the mathematical and graphical data used to check the overlapping of various flights in a unanimous flying pattern so as to neglect causalities and deviation of flights:

7.) If you’ve ever used Google, you’re looking at the world’s most (financially) valuable graph theory application. At the heart of their search engine technology is an algorithm called PageRank, which uses numerous graph theory concepts — including cliques and a lot of connectivity information — to determine how important a given web page is. It does this, in essence, by starting with a rough notion of each page’s importance and then repeatedly refining its estimates by ‘flowing’ importance values from page to page.

8.) Relational Database They play an important part in almost every organization that keep track of its employees, clients or resources. A relational database helps to join a different piece of information. This is all done with the concept of sets in discrete math. Sets allow the info to be grouped and put together. For example: A database containing client information; link the client name, address, phone number, and other info.